Following this trend, in this paper, we explore more deeper properties of discretized sum-product type problems with an emphasis on energy variants and applications. Our work is motivated by earlier results on the \(A+cB\)  problem due to Bourgain in [1]} and Orponen in [2]}, [3]}, [4]}. Let us first start with the following theorem of Bourgain in [1]}. Recall that a set \(A\subset {\mathbb {R}}\)  is called \(\delta \) -separated if every two elements in \(A\)  have distance greater than \(\delta \) .
